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A322104 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{d|n} d*sigma_k(d).

%I A322104 #13 Jan 07 2024 15:52:03
%S A322104 1,1,5,1,7,7,1,11,13,17,1,19,31,35,11,1,35,85,95,31,35,1,67,247,311,
%T A322104 131,91,15,1,131,733,1127,631,341,57,49,1,259,2191,4295,3131,1615,351,
%U A322104 155,34,1,515,6565,16775,15631,8645,2409,775,130,55,1,1027,19687,66311,78131,49111,16815,4991,850,217,23
%N A322104 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{d|n} d*sigma_k(d).
%H A322104 Andrew Howroyd, <a href="/A322104/b322104.txt">Table of n, a(n) for n = 1..1275</a> (first 50 antidiagonals)
%F A322104 G.f. of column k: Sum_{j>=1} j*sigma_k(j)*x^j/(1 - x^j).
%F A322104 L.g.f. of column k: -log(Product_{j>=1} (1 - x^j)^sigma_k(j)).
%F A322104 A(n,k) = Sum_{d|n} d^(k+1)*sigma_1(n/d).
%e A322104 Square array begins:
%e A322104    1,   1,    1,     1,     1,      1,  ...
%e A322104    5,   7,   11,    19,    35,     67,  ...
%e A322104    7,  13,   31,    85,   247,    733,  ...
%e A322104   17,  35,   95,   311,  1127,   4295,  ...
%e A322104   11,  31,  131,   631,  3131,  15631,  ...
%e A322104   35,  91,  341,  1615,  8645,  49111,  ...
%t A322104 Table[Function[k, Sum[d DivisorSigma[k, d], {d, Divisors[n]}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten
%t A322104 Table[Function[k, SeriesCoefficient[Sum[j DivisorSigma[k, j] x^j/(1 - x^j), {j, 1, n}], {x, 0, n}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten
%o A322104 (PARI) T(n,k)={sumdiv(n, d, d^(k+1)*sigma(n/d))}
%o A322104 for(n=1, 10, for(k=0, 8, print1(T(n, k), ", ")); print); \\ _Andrew Howroyd_, Nov 26 2018
%Y A322104 Columns k=0..3 give A060640, A001001, A027847, A027848.
%Y A322104 Cf. A109974, A320940 (diagonal), A321876, A322103.
%K A322104 nonn,tabl
%O A322104 1,3
%A A322104 _Ilya Gutkovskiy_, Nov 26 2018